This book contains the material developed in the framework of the Franco-Indian Spring School held at the University of Saint-Etienne in June 2007 on spectral approximation.The Laboratoire de Mathematique de l'Universite ¿ de Saint-Etienne (LaMUSE, EA 3989) started in 1990 a fruitful scientific collaboration with India and in particular with the Indian Institute of Technology Bombay at Mumbai. Thanks to the programme ARCUS Inde (Actions en Regions de Cooperation Universitaire Scientifique) this cooperation was reinforced and this spring school was organized. This programme was jointly supported by Region Rhone-Alpes and the State Department of Foreign Affairs of France. This spring school allowed us to put together French and Indian specialists working on several aspects of spectral approximation. The goal of spectral approximation is to provide the theoretical and numerical tools for getting information about spectral elements (such as eigenvalues and maximal invariant subspaces) of linear operators on Banach spaces. The conference intended to present sufficiently many different aspects of spectral approximation.